Wave maps from circle to Riemannian manifold: global controllability is equivalent to homotopy

We study wave maps from the circle to a general compact Riemannian manifold. We prove that the global controllability of this geometric equation is characterized precisely by the homotopy class of the data, thereby resolving the conjecture posed in [14, 35]. As a remarkable intermediate result, we establish uniform-time global controllability between steady states, providing a partial answer to an open problem raised in [22]. Finally, we obtain quantitative exponential stability around closed geodesics with negative sectional curvature. This work highlights the rich interplay between partial differential equations, differential geometry, and control theory.